Skew-Adjoint Maps and Quadratic Lie Algebras

The procedure of double extension of vector spaces endowed with non-degenerate bilinear forms allows us to introduce the class of generalized \(\mathbb {K}\)-oscillator algebras over an arbitrary field \(\mathbb {K}\). Starting from basic structural properties and the canonical forms of skew-adjoint endomorphisms, we will proceed to classify the subclass of quadratic nilpotent algebras and characterize those algebras with quadratic dimension 2. This will enable us to recover the classification of real oscillator algebras, a.k.a Lorentzian algebras, given by Medina et al. (Ann Sci École Norm Sup 18:553–561, 1985).